Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure

Abstract

Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $\varphi_\lambda: \Delta_M \varphi_\lambda + \lambda \varphi_\lambda = 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $\lambda$. We show that
$$H^{n-1}(\{ \varphi_\lambda=0\}) \leq C \lambda^{\alpha},$$
where $\alpha>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from above for the volume of the nodal sets in terms of the frequency and the doubling index.

Authors

Alexander Logunov

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
and
Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O.,29B, Saint Petersburg 199178, Russia